Research Article | Open Access
Stable Zero Lagrange Duality for DC Conic Programming
We consider the problems of minimizing a DC function under a cone-convex constraint and a set constraint. By using the infimal convolution of the conjugate functions, we present a new constraint qualification which completely characterizes the Farkas-type lemma and the stable zero Lagrange duality gap property for DC conical programming problems in locally convex spaces.
Let and be real locally convex Hausdorff topological vector spaces and be a nonempty convex set. Let be a closed convex cone and the positive dual cone of . Let be a proper function and be an -convex mapping with respect to the cone . Consider the conic programming problem Its Lagrange dual problem can be expressed as
It is well know that the optimal values of these problems, , and respectively, satisfy the so-called weak duality, that is, , but a duality gap may occur, that is, we may have . A challenge in convex analysis has been to give sufficient conditions which guarantee the strong duality, that is, and the dual problem has at least an optimal solution. In the case when is a proper convex function, numerous conditions have been given in the literature ensuring the strong duality (see, e.g., [1–8] and the other references therein).
Recently, the zero duality, that is, only the situation when , has received much attention (e.g., see [9–13] and references therein). Obviously, the strong duality implies the zero duality. However, the converse implication does not always hold. As mentioned in , the question of finding condition, which ensures the zero duality, is not only important for understanding the fundamental feature of convex programming but also for the efficient development of numerical schemes. Some sufficient conditions and characterizations in terms of the optimal value function of for the zero duality have been given in , and some convex programming problems which enjoy zero duality have been studied in . Especially, in the case when is lower semicontinuous (lsc in brief) convex, is star lsc and is closed; Jeyakumar and Li in  presented some constraint qualifications which completely characterize the zero duality for convex programming problems in Banach spaces; they established necessary and sufficient dual conditions for the stable zero duality in  under the assumptions that and is continuous.
Observe that most works dealing with problem (1.1) in the literature mentioned above were done under the assumptions that the involved functions are convex and lsc. In this paper, we consider the following DC conical programming: and its dual problem where are proper convex functions. As pointed out in , problems of DC programming are highly important from both viewpoints of optimization theory and applications, and they have been extensively studied in the literature (cf. [14–21] and the references therein). Here and throughout the whole paper, following [22, page 39], we adopt the convention that , and . Then, for any two proper convex functions , we have that hence,
The purpose of this paper is to study the stable zero duality. Our main contribution is to provide complete characterizations for the stable zero duality between and via the newly constraint qualifications. In general, we only assume that are proper convex and is -convex (not necessarily lsc).
The paper is organized as follows. The next section contains some necessary notations and preliminary results. The Farkas-type lemma and the stable zero duality between and are considered in Section 3.
2. Notations and Preliminary Results
The notation used in the present paper is standard (cf. ). In particular, we assume throughout the whole paper that and are real locally convex Hausdorff topological vector spaces, and let denote the dual space, endowed with the -topology . By , we will denote the value of the functional at , that is, . Let be a set in . The closure of is denoted by . If , then denotes the -closure of . For the whole paper, we endow with the product topology of and the usual Euclidean topology.
The indicator function of the nonempty set is defined by Let be a proper convex function. The effective domain, the conjugate function, and the epigraph of are denoted by , and , respectively; they are defined by It is well known and easy to verify that is -closed. The lsc hull of , denoted by , is defined by Then (cf. [22, Theorems 2.3.1]), By definition, the Young-Fenchel inequality below holds: If are proper, then Moreover, if is convex and lsc on , then the same argument for the proof of [21, Lemma 2.3] shows that Furthermore, we define the infimal convolution of and as the function given by If and are lsc and , then by , we have that Moreover, we also have Note that an element can be naturally regarded as a function on in such a way that Thus, the following facts are clear for any and any function :
Lemma 2.1. Let be proper convex functions satisfying .(i)If are lsc, then (ii)If either or is continuous at some point of , then
3. Characterizations for the Stable Zero Duality
Throughout this section, let be locally convex spaces and be a nonempty convex set. Let be a closed convex cone. Its dual cone is defined by Define an order on by saying that if . We attach a greatest element with respect to and denote . The following operations are defined on : for any , and for any . Let be proper convex functions such that and are proper, and be -convex in the sense that for every and every , (see ). Let and let . As in , we define for each , It is easy to see that is -convex if and only if is a convex function for each . Following , we define the function by Let . Recall from [19, 23] that is said to be star lsc if is lsc on for each and to be -epi-closed if is closed, where It is known (cf. ) that if is star lsc, then it is -epi-closed. Let denote the solution set of the system , that is, To avoid trivially, we always assume that .
The following lemma, which is taken from [10, Theorem 3.1], will be useful in our study.
Lemma 3.1. Suppose that is a proper star lsc and -convex mapping with . Then(i) is a proper convex function on .(ii) is a convex cone.(iii) and .
Let . Consider the primal problem and its dual problem of In the case when , problem and its dual problem are reduced to problem and its dual problem defined in (1.1) and (1.2), respectively. Let and denote the optimal values of and , respectively. Let , then by the definition of conjugate function, one has that Moreover, in the case when is lsc, then for each , thus, it is easy to see that the following inequality holds: that is, the stable weak Lagrange duality holds. However, (3.11) does not necessarily hold in general as showed in the following example.
Example 3.2. Let and . Define by , and for each , (note that is not lsc at ). Then and . Note that for each , Then . This means that (3.11) does not hold.
Below we give a sufficient condition to ensure that (3.11) holds.
Lemma 3.3. Suppose that the following condition holds: Then (3.11) holds.
Proof. Let . Then for each and , one has by (2.5) that for each , where the last inequality holds because for each . Note that the above inequalities hold for each . Then for each , where the last equality holds by (3.10). Hence, which implies that and by (3.14). Hence, by (3.9), one has . Therefore, (3.11) holds by the arbitrary of . The proof is complete.
This section is devoted to the study of the zero dualities between and , which is defined as follows.
Definition 3.5. We say that(a)the zero duality holds between and if ;(b)the stable zero duality holds between and if for each , the zero duality holds between and .
Definition 3.6. We say the family satisfies the constraint qualification if
The following proposition provides an equivalent condition for to hold.
Proof. To show the equivalence of and (3.20), we only need to show that To do this, by (3.19) and the fact (2.11), it is easy to see that the following inclusion holds: Note that is closed and so is lsc. Then by Lemma 3.1(c), one has that where the last inclusion holds by Lemma 2.1(i). Therefore, where the first equality holds by (2.8) and the last equality holds by (3.14). Hence, (3.21) holds and the proof is complete.
Below we give another sufficient conditions ensuring . For the study of the Lagrange duality and the Fenchel-Lagrange duality, the authors in  introduced the following condition: This condition was also introduced independently but with different terminologies “" and “" in , under the assumptions and [19, 20] (under the assumptions (3.26) together with the star lsc of ), respectively.
Proof. Suppose that holds. Note by the definition that for each . Then and Hence, by (2.11), one has that This together with the implies that Thus, by (2.8) and (3.14), we can obtain that Hence, by Proposition 3.7, holds and the proof is complete.
The converse of Proposition 3.8 does not necessarily hold, even in the case when , as showed in the following example.
Example 3.9. Let and . Let be defined by , and for each . Then and . Hence, . Moreover, it is easy to see that for each , Then, . This implies that . Hence, This means that holds (noting that ). However, note that Then and so the does not hold.
Proposition 3.10. Let . Suppose that (3.19) holds. Then the family satisfies if and only if is -closed.
Proof. Since is lsc and is closed, it follows from Lemma 2.1(i) that while the last equality holds by Lemma 3.1(c). Note by (2.11) that Hence, by (3.36), one has that This together with (3.23) implies that Thus, the result is seen to hold.
The following theorem provides a Farkas-type lemma for the DC optimization problem (3.7) in terms of the condition .
Theorem 3.11. Let and . Suppose that the family satisfies . Consider the following statements.(i)For each , .(ii). (iii)For each and for each , there exists such that Then (i) (ii) (iii). Furthermore, if (3.14) holds, then .
Consider (i)(ii). Suppose that (i) holds. Then for each , . Thus, by (2.7) and the assumed ,
while by (2.14), one has that
Hence, (ii) holds.
Consider (ii)(iii). Suppose that (ii) holds. Let be arbitrary. Then, that is, This means that for each , there exist such that Moreover, by the definition of the function , there exists such that Hence, Therefore, by the Young-Fenchel inequality (2.5), one sees that for each , Note that the above inequalities and the equality hold for each , it follows that Hence, (ii) holds.
Furthermore, suppose that (3.14) holds. Then the weak duality holds between and , that is, . Below we show that (iii)(i). To do this, assume that (iii) holds. Then by the definition of , one has that and by the arbitrary of . Thus, by the weak duality holds between and , one has that . Hence, (i) holds and the proof is complete.
Let denote the set of all points at which is continuous, that is, The following theorem shows that the condition is equivalent to the stable zero duality.
Theorem 3.12. Suppose that (3.14) holds. Consider the following statements.(i)The family satisfies .(ii)The stable zero duality holds between and .Then (i) (ii). Furthermore, (i) (ii) if (3.19) holds and one of the following conditions holds:(a) and ;(b).
Consider (i) (ii). Suppose that (i) holds. Let . If , then the stable zero duality holds between and trivially. Below we assume that . Then by the implication (i)(ii) of Theorem 3.11, one has that . Hence, and, by Lemma 3.3, . Thus, (ii) holds.
Furthermore, suppose that (3.19) holds and one of the conditions (a) and (b) holds. Then, by Lemma 2.1(b), one has that To show (i), by Proposition 3.7, it suffices to show that (3.20) holds. To do this, let . Then, by (3.9), and hence by the stable zero duality between and . Let and , then there exists such that This implies that . Hence, and by the arbitrary of , one has that where the equality holds by (3.51). This together with (3.29) and (2.11) implies that that is, Hence, by the arbitrary of , we have that Therefore, (3.20) holds and the proof is complete.
Recall that in the case when and , under the assumptions that is lsc and is continuous, the authors establish in [11, Theorem 3.1] the equivalence between the stable zero duality and the following regularity condition: In this case, by Proposition 3.10, the following equivalence holds: Hence, the following corollary, which follows from Theorem 3.12, improves the result in [11, Theorem 3.1].
Corollary 3.13. Suppose that Consider the following statements.(i)The family satisfies , that is, (ii)For each , Then . Furthermore, if (3.19) holds and , then .
In the case when , the authors introduce in  the following condition: to study the zero duality between and . Under the assumptions that (3.19) holds and , the authors in  establish the zero duality using the regularity condition . In this case, by Lemma 2.1(b) and Lemma 3.1, we have that This together with Proposition 3.7 implies that By Theorem 3.12, we get the following corollary straightforwardly, which improves the corresponding result in [10, Theorem 4.1], since we do not need to assume that (3.19) holds and .
Corollary 3.14. Suppose that the family satisfies . Then the zero duality holds between and .
Corollary 3.15. Suppose that is closed, is star lsc and that . Then the following statements are equivalent.(i)The condition holds.(ii)If the proper lsc convex function is such that then (iii) If the proper lsc convex function is continuous at some point in , then (3.65) holds.(iv) If , then
Consider (i)(ii). Suppose that (i) holds and let be such that (3.64) is satisfied. Then, it follows from Lemma 3.1(c) that
where the second equality holds by the condition and the last inclusion holds by (2.11). Hence, by Proposition 3.7(a) (note that ), the holds. Applying Corollary 3.14 to in place of , we complete the proof of the implication (i)(ii).
Consider (ii)(iii). Note that (3.64) is satisfied if is continuous at some point in (see Lemma 2.1(ii)). Thus, it is immediate that (ii)(iii).
Consider (iii)(iv). It is trivial.
Consider (iv)(i). Suppose that (iv) holds. Then applying Theorem 3.12 to , one has that Hence, by Lemma 3.1(c), we obtain that that is, the holds.
Using the same argument, one can obtain a sufficient and necessary condition to ensure the zero duality between the primal problem and the Fenchel-Lagrange duality.
Theorem 3.16. Suppose that (3.14) holds. Consider the following statements.(i)The family satisfies the following condition: (ii)For , the following equality holds: Then (i) (ii). Furthermore, if (3.19) holds and either or , then .
The author is grateful to the anonymous reviewer for valuable suggestions and remarks helping to improve the quality of the paper. This author was supported in part by the National Natural Science Foundation of China (grant 11101186).
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